Our aim in Section 8 is, motivated by the work of Huneke and Wiegand [1]}, to determine some new criteria for freeness of modules in terms of the vanishing of local cohomology. More precisely, we are concerned with the cohomological rigidity property of tensor products in the sense that the freeness of a module \(M\)  follows from the vanishing of \( \operatorname{H}^{i}_{\mathfrak {m}}(M\otimes _RM^\ast )\)  for some integer \(i\) . Our work extends several results from the literature. For example, as a consequence of Theorem REF , we proved the following result in Corollary REF ; it extends [2]} which establishes the case where \(n=0\) :
